Calculating Length

I was studying for my multi-variable calculus test, when I realized that their seem to be two very similar ways to calculate the length. (used to calculate arc length) vs (used to calculate length of normal vector/ any vector)
And I guess they really are basically the same, but I was wondering if someone could explain the difference, and perhaps why the first one is used in line integrals whereas you use the second in surface integrals.
Thanks!

I was studying for my multi-variable calculus test, when I realized that their seem to be two very similar ways to calculate the length. (used to calculate arc length) vs (used to calculate length of normal vector/ any vector)
And I guess they really are basically the same, but I was wondering if someone could explain the difference, and perhaps why the first one is used in line integrals whereas you use the second in surface integrals.
Thanks!

The difference is pretty clearly the fact that arcs are not (necessarily) linear and vectors are.

The integral formula is a lot of sums of very small lines (you know what I mean). The vectors representing those lines talk about how quickly the function's value is changing. So if you sum up the lengths of those vectors, you get the large scale change of the function.

The reason, then, that the two formulas look so similar is because those vector's lengths were needed to compute the components of the arc length.

I highly doubt there is any deep reason one tends to be line integrals and the other surface, but keep in mind that one is measuring things about a curve and the other could find something about a vector normal to the surface, which as you know from linear surfaces tend to reveal a lot about the surface itself.